A Finite-Peeling Proof of the Collatz Conjecture via the Directed ABN Baumkuchen Representation

This paper uses alternating binary notation (ABN) and Collatz phase expression (CPE) to describe the accelerated Collatz map on positive odd integers as an operation on finite words, and formulates an ``ABN Baumkuchen representation'' that transfers this operation to a finite sequence of directed reserved rings. Ordinary ABN zero digits, unused reserved positions, and a special symbol marking a ring cut are separated as distinct types. The special cut is not a numerical digit; it only marks the connection through which the displayed upper carry line is routed to the unique virtual $+1$ input at the inlet of the active lowest ring.

The initial canonical ABN is wound onto a single active lowest ring, with finitely many empty reserved rings placed above it. Only the lowest ring receives the Collatz-specific $+1$ input and division by two. Every upper ring receives only a directed carry from the ring immediately below. Although a boundary-crossing arithmetic carry is a single carry state, local canonicalization in the receiving ring may expand it into a finite contiguous ABN block containing several digits. That block is inserted consecutively from the inlet, and only the finite remainder exceeding the reserved width is passed to the next ring. Thus the total number of effective ABN positions may increase, but information leaving the active lowest ring never returns to refill that ring, and division by two strictly decreases its occupied length.

When the lowest ring becomes empty, an unfilled next ring can be unrolled to a canonical ABN shorter than the original one, whereas a filled next ring inherits the role of active lowest ring. A finite reservation sequence with strictly decreasing widths down to width two therefore reaches, after finitely many steps, either a shorter canonical ABN or a word of the form $(pm)^r$. The latter reaches $1$ in one accelerated Collatz step. This terminal dichotomy is then combined with strong induction on canonical ABN length.